AS Mathematics Assignment 7 Due Date: Friday 14 th February 2014

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1 AS Mathematics Assignment 7 Due Date: Friday 14 th February 2014 NAME. GROUP: MECHANICS/STATS Instructions to Students All questions must be attempted. You should present your solutions on file paper and submit them with this cover sheet. (Work submitted without a cover sheet complete with name will not be marked). All work is to be submitted either to your module teacher or the Faculty Office by 4.1pm on the due date above. SKILLS AUDIT PLEASE COMPLETE BELOW, YOUR TEACHER TO CONFIRM THE SKILLS MET IN THIS ASSIGNMENT SKILLS AUDIT YES NO NEEDS WORK I can find the nth term of a geometric progression I can find the sum of the 1 st n terms of a geometric progression I can find the sum to infinity of terms of a geometric progression I can construct an algebraic expression from information given about consecutive terms in a geometric progression I can solve geometric progression problems when given information about the terms in the sequence I can find the number of terms in a finite geometric progression using trial and improvement or logarithms I can solve problems involving growth and decay REFERENCES Core Maths 2 Edexcel text book Geometric sequences: pages Geometric progressions and the nth term: pages Using geometric sequences to solve problems: pages The sum of a geometric series: pages The sum to infinity of a geometric series: pages Students Feedback: Please comment on how you performed on this assignment and state the questions that you found difficult. TURN OVER FOR COMMENTS ABOUT YOUR WORK Haringey Sixth Form Centre Mathematics Department 1

2 TEACHER COMMENTS Strengths Areas to develop Haringey Sixth Form Centre Mathematics Department 2

3 1. The fourth term of a geometric series is 10 and the seventh term of the series is 80. For this series, find the common ratio, the first term, the sum of the first 20 terms, giving your answer to the nearest whole number. (Total 6 marks) 2. The second and fifth terms of a geometric series are 9 and 1.12 respectively. For this series find the value of the common ratio, the first term, the sum to infinity. (Total 7 marks) 3. The second and fourth terms of a geometric series are 7.2 and.832 respectively. The common ratio of the series is positive. For this series, find the common ratio, the first term, the sum of the first 0 terms, giving your answer to 3 decimal places, the difference between the sum to infinity and the sum of the first 0 terms, giving your answer to 3 decimal places. (Total 8 marks) 4. A car was purchased for on 1st January. On 1st January each following year, the value of the car is 80% of its value on 1st January in the previous year. Show that the value of the car exactly 3 years after it was purchased is The value of the car falls below 1000 for the first time n years after it was purchased. Find the value of n. Haringey Sixth Form Centre Mathematics Department 3

4 An insurance company has a scheme to cover the maintenance of the car. The cost is 200 for the first year, and for every following year the cost increases by 12% so that for the 3rd year the cost of the scheme is Find the cost of the scheme for the th year, giving your answer to the nearest penny. Find the total cost of the insurance scheme for the first 1 years.. A trading company made a profit of in 2006 (Year 1). A model for future trading predicts that profits will increase year by year in a geometric sequence with common ratio r, r > 1. The model therefore predicts that in 2007 (Year 2) a profit of 0 000r will be made. Write down an expression for the predicted profit in Year n. The model predicts that in Year n, the profit made will exceed log 4 Show that n > 1. log r Using the model with r = 1.09, find the year in which the profit made will first exceed , find the total of the profits that will be made by the company over the 10 years from 2006 to 201 inclusive, giving your answer to the nearest A geometric series has first term and common ratio. Calculate the 20th term of the series, to 3 decimal places, the sum to infinity of the series. Given that the sum to k terms of the series is greater than 24.9, log0.002 show that k, log0.8 find the smallest possible value of k. Haringey Sixth Form Centre Mathematics Department 4

5 7. The third term of a geometric sequence is 324 and the sixth term is 96 Show that the common ratio of the sequence is Find the first term of the sequence. Find the sum of the first 1 terms of the sequence. Find the sum to infinity of the sequence The first three terms of a geometric series are (k + 4), k and (2k 1) respectively, where k is a positive constant. Show that k 2 7k 60 = 0. Hence show that k = 12. Find the common ratio of this series. Find the sum to infinity of this series. (Total 10 marks) 9. A geometric series is a + ar + ar Prove that the sum of the first n terms of this series is given by S n n 1 r. a 1 r Find 10 k k. Find the sum to infinity of the geometric series State the condition for an infinite geometric series with common ratio r to be convergent. (Total 11 marks) Haringey Sixth Form Centre Mathematics Department

6 10. A geometric series has first term a and common ratio r. The second term of the series is 4 and the sum to infinity of the series is 2. Show that 2r 2 2r + 4 = 0. Find the two possible values of r. Find the corresponding two possible values of a. Show that the sum, S n, of the first n terms of the series is given by S n = 2(1 r n ). Given that r takes the larger of its two possible values, (e) find the smallest value of n for which S n exceeds 24. (Total 11 marks) 11. The first term of a geometric series is 120. The sum to infinity of the series is 480. Show that the common ratio, r, is 3. 4 Find, to 2 decimal places, the difference between the th and 6th term. Calculate the sum of the first 7 terms. The sum of the first n terms of the series is greater than 300. Calculate the smallest possible value of n. (Total 11 marks) Haringey Sixth Form Centre Mathematics Department 6

7 STRETCH AND CHALLENGE QUESTION! 12. The adult population of a town is at the end of Year 1. A model predicts that the adult population of the town will increase by 3% each year, forming a geometric sequence. Show that the predicted adult population at the end of Year 2 is Write down the common ratio of the geometric sequence. The model predicts that Year N will be the first year in which the adult population of the town exceeds Show that (N 1) log1.03 > log1.6 Find the value of N. At the end of each year, each member of the adult population of the town will give 1 to a charity fund. Assuming the population model, (e) find the total amount that will be given to the charity fund for the 10 years from the end of Year 1 to the end of Year 10, giving your answer to the nearest (Total 10 marks) END OF ASSIGNMENT Haringey Sixth Form Centre Mathematics Department 7